Score Tests in Generalized Linear Measurement Error Models

SUMMARY Hypothesis tests in generalized linear models are studied under the condition that a surrogate w is observed in place of the true predictor x. The efficient score test for the hypothesis of no association depends on the conditional expectation E(xI w) which is generally unknown. The usual test substitutes w for E(xI w) and is asymptotically valid but not efficient. We investigate two new test statistics appropriate when w = x + z where z is an independent measurement error. The first is a Wald test based on estimators corrected for measurement error. Despite the correction for attenuation in the estimator, this test has the same local power as the usual test. The second test employs an estimator of E(x I w) and is both asymptotically efficient for normal errors and approximately efficient when the measurement error variance is small.